1) Determine the equation of the carcle a) Write down the value radus 5 3) Determine the gradient of the radus 5

Question

UpStudy AI · Accepted Answer

To solve the problem, let's break it down step by step.

### Step 1: Determine the equation of the circle

The standard equation of a circle with center at the point $$(h, k)$$ and radius $$r$$ is given by:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

In this case, you mentioned a radius of 5. However, we need the center of the circle to write the complete equation. If we assume the center is at the origin $$(0, 0)$$, the equation becomes:

$$
x^2 + y^2 = 5^2
$$

This simplifies to:

$$
x^2 + y^2 = 25
$$

### Step 2: Write down the value of the radius

The radius $$r$$ is given as 5.

### Step 3: Determine the gradient of the radius

The gradient (or slope) of a line is calculated as the change in $$y$$ over the change in $$x$$. For a radius of the circle, we can consider a point on the circle. The coordinates of a point on the circle can be expressed in terms of the angle $$\theta$$ as:

$$
(x, y) = (5 \cos(\theta), 5 \sin(\theta))
$$

To find the gradient of the radius at this point, we can differentiate $$y$$ with respect to $$x$$:

1. Differentiate $$y = 5 \sin(\theta)$$ with respect to $$\theta$$:
   $$
   \frac{dy}{d\theta} = 5 \cos(\theta)
   $$

2. Differentiate $$x = 5 \cos(\theta)$$ with respect to $$\theta$$:
   $$
   \frac{dx}{d\theta} = -5 \sin(\theta)
   $$

3. The gradient $$m$$ is given by:
   $$
   m = \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{5 \cos(\theta)}{-5 \sin(\theta)} = -\cot(\theta)
   $$

Thus, the gradient of the radius at any point on the circle is $$-\cot(\theta)$$, where $$\theta$$ is the angle corresponding to the point on the circle.

### Summary

1. The equation of the circle is $$x^2 + y^2 = 25$$.
2. The radius is 5.
3. The gradient of the radius at a point on the circle is $$-\cot(\theta)$$.
The equation of the circle is $$x^2 + y^2 = 25$$, the radius is 5, and the gradient of the radius at any point on the circle is $$-\cot(\theta)$$.

1) Determine the equation of the carcle a) Write down the value radus 5 3) Determine the gradient of the radus 5

Solución de tutoría real

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Geometry Questions